On efficient abstract method for the study of Cauchy problem for a second order differential equation set on a singular cylindrical domain

(1)

On efficient abstract method for the study of Cauchy

problem for a second order differential equation set

on a singular cylindrical domain

Belkacem Chaouchi

1

*

Abstract

In this work, we present an abstract approach for the study of a mixed initial value problem set in cylindrical domain with a cusp base. MaximalLpregularity results for the strict solution are established.

Keywords

Fractional powers of linear operators; analytic semigroup, Abstract differential equation, Cuspidal point.

AMS Subject Classification

34G10, 34K10, 12H2O, 44A45.

1_{Lab. de l’Energie et des Syst `emes Intelligents, Khemis Miliana University, Algeria.}

*Corresponding author:[email protected]

Article History: Received12October2017; Accepted05February2018 c2018 MJM.

1. Introduction

The regularity analysis for boundary value problems set on sin-gular domains has attracted the attention of many researchers. The solvability of such problems was discussed by means of different methods such as the variational methods or the well known potential theory. For more information, we can refer the reader to [13], [14], [15] and the references cited therein. In this work, we consider the cusp domainΠ⊂R3defined by

Π=

(x,y,z)∈_R3_{: 0}_<_z_<₁_,

_x

zα,

y zα

∈D

, α>1,

(1.1)

where

D=

(x,y)∈_R2_{: 0}_<_x2₊_y2_<₁ _.

We want to study the problem

∂_t2u+∆u=h, on]0,1[×Π, (1.2)

associated to the following mixed boundary conditions

∂t2u

_{₀_}×D+∆_(x_,_y)u _{

1}×D=u0,

∂t2u

_{₁_}×D−∆_(x_,_y)u _{

0}×D=u1, u|_]₀_,₁_[×_∂_D=0.

(1.3)

Here,∆stands for the standard Laplacian inR3andu0,u1are

given functions.

We assume that the second memberhis taken in the Lebesgue

spacesLp(0,1;Lp(Π)),3

2 <p<∞. Note here that the bound-ary conditions (1.3) are a considered as restriction of the following ones

∂_t2u_{

0}×D+∆u|{1}×D=u0, ∂_t2u_{

1}×D+∆u|{0}×D=u1,

u|_]₀_,₁_[×_∂_D=0.

(1.4)

(2)

of a forthcoming study. In our situation, the boundary of the domainΠgiven by (1.1) is at the origin of several difficulties which make the use of the classical tools a complicated task. Then, we have opted for the use of another approach based essentially on the theory of abstract differential equations. The effectiveness of this method was manifested in several works, see for example [2], [3], [5], [7], [8], [18] and the references cited therein. Our purpose is to establish existence, uniqueness and maximal regularity of the strict solution for (1.2)-(1.3). For this reason, some suitable compatibility con-ditions onu0andu1will be imposed.

We will prove that the study of our problem can be reduced to the study of an abstract differential equation associated to some nonlocal boundary conditions involving a linear un-bounded operator. The techniques of investigation used here are based on the use of analytic semigroup’s approach com-bined with the sum’s operators theory developed in [6]. This work is organized as follows, in the next section, we show that our problem can be transformed by a natural changes of variables into an abstract second order differential problem. Section 3, is devoted to the study of the abstract version of the transformed problem. Finally, in section 4, we go back to our first problem in the non regular cylindrical domain and we justify our main results.

2. The abstract setting of the problem

2.1 Change of variables

First, we use the following change of variables

T: ]0,1[×Π→]0,1[×Ω,

(t,x,y,z)7→(t,ξ,η,λ), λ>λ0 (2.1)

where

    

   

(t,ξ,η,λ) =

t,_zxα, y zα,

θ

zβ

Ω=D×]λ₀,+∞[,

λ0=_α1₋₁>0,

β =α−1=1

θ.

Putting

v(t,ξ,η,λ) =u(t,x,y,z)andg(t,ξ,η,λ) =h(t,x,y,z).

Then, equation (1.2) is transformed into a new one given by

∂_t2v−

θ

λ −2α/β

∆v+

θ

λ −2α/β

Pv=g, (2.2) with

Pv(t,ξ,η,λ)

= (

1+α2θ2

ξ

λ

2)

∂_ξ2v+

1+α2θ2

_η

λ

2

∂_η2v

+2α2θ2ξ η

₁

λ 2

∂_{ξ η}2 v+2α θ

ξ

λ

∂_{ξ λ}2 v (2.3)

+2α θ

ξ2

λ

∂_{η λ}2 v+α(α+1)θ2ξ

₁

λ 2

∂ξv

+α(α+1)θ2η

₁

λ 2

∂ηv

+α θ

1

λ

∂λv.

Now, let us introduce the following change of functions

  

 

v= θ λ

s

w,

and

f = ((α−1)λ)

−3α p(α−1)_g_,

with

s=− α

α−1

₃

p−2

=−α

β

₃

p−2

.

Consequently, equation (2.2) becomes

k(λ)∂_t2w−∆w+1

λLw=f, (2.4)

with

k(λ) =

λ

θ s

,

and

Lw(t,ξ,η,λ)

= (α θ) 2

λ n

ξ2∂_ξ2w+η2∂_η2w+2ξ η ∂_{ξ η}2 w

o

+2α θ n

ξ ∂_{ξ λ}2 w+ξ2∂_{η λ}2 w o

+ (α θ−2s)∂_λw

+α θ

λ ((α+1)θ−2s)

ξ ∂ξw+η ∂ηw

+s

λ(s+1−α θ)w.

Concerning the boundary conditions, it is easy to see that (1.3) are transformed to

k(λ)∂_t2w(1,ξ,η,λ)−∆(ξ,η)w(0,ξ,η,λ) =k(λ)w0,

k(λ)∂_t2w(0,ξ,η,λ) +∆₍_ξ_,_η₎w(0,ξ,η,λ) =k(λ)w1,

w(t,ξ,η,λ) =0, (t,ξ,η,λ)∈]0,1[×∂Ω,

(3)

2.2 Statement of the abstract problem

Without loss of generality, we consider the following simpli-fied problem

∂t2w−∆w=f, (2.5)

under the corresponding conditions

∂_t2w(1,ξ,η,λ)−∆(ξ,η)w(0,ξ,η,λ) =w0, (ξ,η,λ)∈Ω,

∂_t2w(0,ξ,η,λ) +∆(ξ,η)w(0,ξ,η,λ) =w1, (ξ,η,λ)∈Ω, w(t,ξ,η,λ) =0, (t,ξ,η,λ)∈]0,1[×∂Ω.

(2.6)

Now, setE=Lp(Ω)andX=Lp(0,1;E)with3

2 <p<∞and consider the following vector-valued functions

w : ]0,1[→E;t−→w(t); w(t) (ξ,η,λ) =w(t,ξ,η,λ),

f : ]0,1[→E;t−→ f(t); f(t) (ξ,η,λ) =f(t,ξ,η,λ).

Set

 



D(M) =W2,p(Ω)∩W₀1,p(Ω),

Mψ=∆ψ, ψ∈D(M).

 



D(N) =W2,p(D)∩W₀1,p(D),

Nψ=∆₍_ξ_,_η₎ψ,ψ∈D(N).

and define the following operators.

 



D(A) ={ψ∈X :ψ(t)∈D(M), a.et∈]0,1[},

(Aψ)(t) =M(ψ(t)), t∈]0,1[,

(2.7)

 



D(H) ={ψ∈X :ψ(t)∈D(N), a.et∈]0,1[},

(Hψ)(t) =N(ψ(t)), t∈]0,1[.

(2.8)

Consequently, the abstract version of (2.5)-(2.6) is given by

w00(t) +Aw(t) =f(t), t∈]0,1[, (2.9)

w00(1)−Hw(0) =w1, w00(0) +Hw(1) =w0, (2.10)

wherew0,w1are given elements of the complex Banach space

E.Our purpose is to find a strict solution of Problem (2.9)-(2.10), i.e. a functionwsuch that

 



w∈W2,p(0,1;E)∩Lp(0,1;D(A)),

w0,w1∈D(H),

wsatisfies (2.10).

In the sequel, we need the following results describing some spectral properties of the above cited operators.

Definition 2.1. Let E be a complex Banach space and B be a closed linear operator in E, denoting byρ(B)its resolvent set. Then, B is said to be sectorial if there are constantsµ

∈R,δ ∈(π/2,π), M>0such that

   

  

ρ(B)⊃Sδ,µ={z∈C,z6=µ: arg(z−µ)<δ},

(B−zI)−1 _L(E)6

C |z−µ|

, z∈Sδ,µ.

(2.11)

Lemma 2.2. The densely closed linear operator(A,D(A))

defined by (2.7) is a sectorial operator withµ=0andδ =π₂.

Proof. See section 3.1.1 in [16], in where a complete study of more general elliptic operator was discussed.The density ofD(A)follows from Proposition 2.1.1 in [11].

Similarly, one has

Lemma 2.3. The densely closed linear operator(H,D(H))

defined by (2.8) is a sectorial operator withµ=0andδ =π₂.

Remark 2.4. In our situation, observe that

1. it is easy to see that

AH=HA,

2. the estimate (2.11) implies that

Q=−√−A,

is well defined. Furthermore, there exists a sector

Πδ,r0

= {z∈C∗:|argz| ≤δ+π/2} ∪B(0,r0),

(with some positiveδ, r0) and C>0such that

   

  

ρ(Q)⊃Πδ,r0

and

∀z∈Πδ,r0,

(Q−zI)−1≤

C |z|.

Thus, one has for all t∈]0,1[andϕ∈E,

etQϕ= 1

2iπ

Z

γ1

etz(Q−zI)−1ϕdz,

whereγ1is a suitable sectorial curve in the complex

plane.

From Remark 1 in [4], it follows that

Lemma 2.5. For anyϕ∈E,k∈N∗and t∈]0,1[, one has

eQϕ∈D Qk,

eQϕ∈Lp(0,1;E),

and

 



QketQeQϕ∈Lp(0,1;E),

QketQeQϕ=etQQkeQϕ,

(4)

3. Regularity results for the transformed

problem

3.1 Representation of the solution

Using the Krein’s method, we know that the unique solution wof (2.9)-(2.10) is given by

w(t)

= b1etQ+b2e(1−t)Q+I(t) +J(t),

where

I(t) = Q

−1

2 t

Z

0

eQ(t−s)f(s)ds,

J(t) = Q

−1

2

1

Z

t

eQ(s−t)f(s)ds.

The constantb1andb2are uniquely determined via the

bound-ary conditions (2.10). For more informations about this tech-nique, we refer the reader to [1] and the references therein. Using the commutativity of the two operators, namely(A,D(A)), (H,D(H)), we obtain the following abstract system

     

    

Q2_eQ₋_H

b1+ Q2−HeQb2 =w1−Q2I(1) +HJ(0),

Q2+HeQb1+ Q2eQ+H

b2 =w0−Q2J(0)−HI(1).

(3.1)

The abstract determinant of (3.1) is given by

Λ=− 1−e2Q

Q4+H2,

and one has

Lemma 3.1. The operator

Λ

= − 1−e2Q

Q4+H2 ,

is closed and boundedly invertible. Furthermore

Λ−1 (3.2)

= − Q4+H2−1

1−e2Q−1

, (3.3)

Proof. Thanks to Proposition 2.3.6 in [16], we know that the operator 1−e2Q

has a bounded inverse

1−e2Q−1

= 1

2iπ

Z

γ1

g(z)−1

g(z) (Q−zI)

−1_dz₊_I_,

where

g(z) =1−e2z,

andγ1is a suitable curve in the complex plane.

On the other hand, The Dore-Venni sum’s theory allows us to confirm that

Q4+H2,

is closed and Q4+H2−1∈L(E). Furthermore, one has

Q4+H2−1

= − 1

2iπ

Z

γ2

(AH)2zH−2

sinπz dz,

γ2is also a suitable curve in the complex plane, see Theorem

2.1 in [6].

Using the Dunford’s operational calculus, we write

Q4+H2−1

1−e2Q−1

=

₁

2iπ

2 Z

γ1×γ2

(AH)2zH−2 sinπz

g(µ)−1

g(µ) (Q−µI)

−1_dzd µ

− 1 2iπ

Z

Γ2

(AH)2zH−2 sinπz dz

At this level, Fubini’s theorem allows us to write

Q4+H2−1

1−e2Q−1 = 

−

1 2iπ

Z

γ1

g(µ)−1

g(µ) (Q−µI)

−1_d µ





(×) 

−

1 2iπ

Z

γ2





− 1 2iπ

Z

γ2

= 1−e2Q−1

Q4+H2−1 .

Summing up, we deduce that the formal solution of (2.9)-(2.10) is given by the formula

w(t)

= etQΛ−1Hw1−Q2w0

+etQeQΛ−1Hw0+Q2w1

+e(1−t)QΛ−1Hw0+Q2w1 (3.4)

−e(1−t)Q_eQ

Λ−1Hw1−Q2w0

+etQ 1−e2Q−1J(0)

+etQeQ

h

2Λ−1Q4I(1) + 1−e2Q−1I(1)i

+e(1−t)Q 1−e2Q−1[I(1)−J(0)]

(5)

Putting

              

             

d0=Λ−1

Hw1−Q2w0

,

d1=Λ−1Hw0+Q2w1

,

¯

d0= 1−e2Q

−1

J(0),

¯

d1=2Λ−1Q4I(1) + T I(1),

¯

d2= 1−e2Q −1

[I(1)−J(0)].

(3.5)

Then,wis given by the following compact expression

w(t) =G1(t) +G2(t) +G3(t) +G4(t),

with

         

        

G1(t) =etQd0+e(1−t)Qd1+etQd¯0,

G2(t) =etQeQd1−e(1−t)QeQd0,

G3(t) =etQd¯0+etQeQd¯1+e(1−t)Qd¯2,

G4(t) =I(t) +J(t).

3.2 Study of regularity of the solution From [6]), one has

Lemma 3.2. . Let f ∈Lp(0,1;E), 3₂<p<∞. Then, the following applications

t→Q t

R

0

e(t−s)Qf(s)ds,

t→Q

1

R

t

e(s−t)Qf(s)ds,

are well defined for a.e. t∈]0,1[and belong to Lp(0,1;E).

To establish more regularity results, we need to introduce for any closed operatorBandδ ∈]0,1[,the following

clas-sical real interpolation spaces betweenD(B)andEdefined by

(E,D(B))_δ_,_p= (D(B),E)₁₋_δ_,_p

For more details about these spaces, see [9]. In our situation, sinceQgenerates an analytic semigroup, it follows from [17] that beginlemma For all3₂<p<∞andϕ∈E, one has

      

     

Qe.Qϕ

∈Lp(0,1;E)⇔ϕ∈(D(A),E)1 2p+12,p,

and Ae.Qϕ

∈Lp(0,1;E)⇔ϕ∈(D(A),E)1 2p,p

.

Keeping in mind that I(0)and J(1)are bounded and due to the regularizing effect of eQ_, _Λ−1_and _T_,_{the following}

regularity results are a direct consequence of Lemma3.2.

Proposition 3.3. Let d0, d1,d¯0,d¯1,d¯2given by (3.5). Then,

for 3₂<p<∞, one has

1. t → Q2G1(t) ∈ Lp(0,1;E) if and only if

d0, d1∈ D(Q2),E

1 2p,p

.

2. t→Q2G2(t)∈Lp(0,1;E).

3. t→Q2G3(t)∈Lp(0,1;E).

4. t→Q2G4(t)∈Lp(0,1;E).

The above proposition can be equivalently formulated as follows

Theorem 3.4. Let f∈Lp(0,1;E), 3₂<p<∞. and w0, w1∈

E Then, the following assertions are equivalents

1. Hw1,Aw0, Hw0,Aw1∈(D(A),E)1 2p,p

.

2. w given by (3.4) is the unique strict solution of (2.9 )-(2.10) satisfying the maximal regularity property, that is

w00and Aw∈Lp(0,1;E).

In order to state our main results, we recall the definition of the following Besov space forυ∈(0,1)

Bυ

p,p(Ω)

: =  



ψ∈Lp(Ω):

Z

Ω Z

Ω

|ψ(ξ1)−ψ(ξ2)|p

kξ1−ξ2k1+pυ

dξ1dξ2<∞  



whereξ1andξ2are a two generic points ofR3, see [9], p.

680, and one has

Lemma 3.5. Let A the operator defined by (2.7) and 3₂<p<

∞. Then

(D(A),E)1 2p+

1 2,p

=

ψ∈Lp 0,1;Bυ_p(Ω): ψ=0on∂Ω

: =Lp0,1; ˚Bυ

p(Ω)

.

Proof. One has

(D(A),E)1 2p+12,p

=Lp

0,1;W₀2,p(Ω)∩Lp(Ω)

1 2p+12,p

First, it is well known that

Lp

0,1;W₀2,p(Ω)

⊂ Lp

0,1;W₀2,p(Ω)∩Lp(Ω)

1 2p+

1 2,p

⊂ Lp(0,1;Lp(Ω)).

From [9], Proposition 3, P. 683, one has

W2,p(Ω)∩Lp(Ω)1 2p+12,p

=B1−

1

(6)

and as in p. 708 in the same work cited above, we get

W₀2,p(Ω)∩Lp(Ω)

1 2p+

1 2,p

=

ψ∈B 1−1

p

p (Ω):ψ=0 on∂Ω

: =B˚1−

1

p p (Ω)

Then, we can formulate our main results in the trans-formed domain as follows.

Theorem 3.6. Let f∈Lp((0,1)×Ω),3₂<p<∞. Then, the following assertions are equivalents

1. ∆(ξ,η)w1,∆w0,∆(ξ,η)w0,∆w1∈Lp

0,1; ˚B1−

1

p p (Ω)

.

2. Problem (2.5)-(2.6) has a unique strict solution

w∈W2,p(0,1;Lp(Ω))∩Lp(0,1;W₀2,p(Ω)),

satisfying the maximal regularity property, that is

∂_t2w and∆w∈LP(0,1;Lp(Ω)).

Using identically a classical perturbation argument as in [10], we conclude that

Theorem 3.7. Let f ∈Lp((0,1)×Ω), 3₂<p<∞. Then the following assertions are equivalents.

1. ∆(ξ,η)w1,∆w0,∆(ξ,η)w0,∆w1∈Lp

0,1; ˚B1−

1

p p (Π)

.

2. Problem (2.4)-(2.6) has a unique strict solution

w

∈ W2,p(0,1;Lp(Ω))∩Lp(0,1;W₀2,p(Ω)),

satisfying the maximal regularity property, that is

∂_t2w and∆w∈Lp(0,1;Lp(Ω)).

4. Regularity results for problem

(1.2)-(1.3)

From [2], the go back to our original domain use the following inverse change of variables

T−1: ]0,1[×Ω→]0,1[×Π,

(t,ξ,η,λ)7→

t, θ λ

α/β

ξ, θ_λα/βη, θ_λ1/β

.

(4.1)

Since

w =

θ

λ 3α/pβ

z−2α_u

= z3α/p_z−2α_u_,

then, a direct computation leads to

(

∂_ξw= θ λ

3α/pβ

z−α_∂

xu=z3α/pz−α∂xu,

∂ηw= θ_λ

3α/pβ

z−α_∂

yu=z3α/pz−α∂yu,

and

   

  

∂_ξ2w= θ_λ3α/pβ∂_x2u=z3α/p∂_x2u,

∂_η2w= θ λ

3α/pβ

∂_y2u=z3α/p_∂2

yu,

∂_{ξ η}2 w= θ_λ3α/pβ∂_xy2u=z3α/p∂_xy2u.

As an immediate consequence of the preceding maximal reg-ularity results with respect to(t,ξ,η,λ), we deduce then that

z−2α_u_,_z−α

∂xu,z−α

∂yu,∂_xy2u,∂_x2u,∂_y2u

∈ Lp(0,1;Lp(Π)), 3

2 <p<∞

On the other hand, since∂_λ2w=∆w−∆(ξ,η)w, then

∂_z2u∈Lp(0,1;Lp(Π)).

Regarding now the cross derivatives∂_{ξ λ}2 wand∂_{η λ}2 w, one has

∂_{ξ λ}2 w=              

            

θ λ

3α/pβ

α

2α−3α

p

θ λz

−α

∂xu

−αθ_λ ∂_x2u+∂_xy2u−∂_xz2u

,

and

θ λ

3α/pβ

α

2α−3_pα

θ λz

−α_∂_yu

−αθ_λ ∂_y2u+∂_xy2u−∂_xz2u

,

from which, we deduce that

∂xzu,∂yzu∈Lp(0,1;Lp(Π)), 3

2<p<∞.

Finally, one has

∂λw =

θ

λ

3α/pβ

2α−3α

p

θ

λz

−2α_u

−

θ

λ

3α/pβ αθ

λz

−α_(∂

xu+∂yu)−z−α∂zu

,

it follows then that

z−α

∂zu∈Lp(0,1;Lp(Π)), 3

2 <p<∞

Summing up, one has

z−2α_u_,_z−α

∂xu,z−α∂yu,z−α∂zu,

(7)

Similarly, the concrete version of the compatibility conditions onu0andu1are given by

       

      

z−2α_u

0,z−α∂xu0,z−α∂yu0,z−α∂zu0,z∂_xy2u0,∂_x2u0,∂_y2u0

∈B˚2−

1

p,p p,# (Π),

and z−2α_u

1,z−α∂xu1,z−α∂yu1,z−α∂zu1,∂xy2u1,∂_x2u1,∂_y2u1

∈_B˚2−1p,p p,# (Π).

where

˚ B2−1p,p

p,# (Π):=

ψ:z3αp_ψ_∈B˚2−

1

p,p p (Π)

.

Summing up, we are in position to state our main results

Theorem 4.1. Let h∈Lp(0,1;Lp(Π)), 3₂<p<∞.Assume that

       

      

z−2α_u

0,z−α∂xu0,z−α∂yu0,z−α∂zu0,z∂_xy2u0,∂_x2u0,∂_y2u0

∈_B˚2−1p,p p,# (Π),

and z−2α_u

1,z−α∂xu1,z−α∂yu1,z−α∂zu1,∂xy2u1,∂_x2u1,∂_y2u1

∈_B˚2−1p,p p,# (Π).

Then problem (1.2)-(1.3) has unique strict solution

u∈W2,p(0,1;Lp(Π))∩Lp(0,1;W₀2,p(Π)),

such that

     

    

∂_t2u

∈Lp(0,1;Lp(Π)),

and z−2α_u_,_z−α

∂xu,z−α∂yu,z−α∂zu,∂xy2u,∂xzu,∂yzu,∂x2u,∂y2u,∂z2u ∈Lp(0,1;Lp₍_Π_)).

Acknowledgments

The authors would like to express sincere gratitude to the reviewers for his/her valuable suggestions.

References

[1] _{A. Aibeche, N, Amroune , S. Maingot; On Elliptic}

Equa-tions with General Non-Local Boundary CondiEqua-tions in UMD Spaces, Mediterr. J. Math,13, No 3, 1051–1063

[2] _{K. Belahdji; La régularité}_Lp_{de la solution du problème} de Dirichlet dans un domaine à points de rebroussement, C. R. A. S, Sér. I 322, 1996, 5–8.

[3] _{T. Berroug, D. Hua, R. Labbas, B. K Sadallah; On a}

Degenerate Parabolic Problem in H¨older Spaces. Applied Mathematics and Computation, Vol. 162, Issue 2, 2005, 811-833.

[4] _{M. Cheggag, A. Favini, R. Labbas, S. Maingot, A.}

Medeghri; Sturm-Liouville Problems for an Abstract Dif-ferential Equation of elliptic Type in UMD Spaces, Diff. Int. Eq, Vol 21, 2008, 981-1000.

[5] _{M. B. Dhakne,D. Kucche Kishor;} _{Second order}

Volterra-Fredholm functional integrodifferential equa-tions, Malaya J. Mat, Spec. Iss, 1-7 2012.

[6] _{G. Dore and A. Venni; On the Closedness of the Sum of}

two Closed Operators, Mathematische Zeitschrift, 196, 1987, 270-286.

[7] _{Dore G., Venni A; An Operational Method to Solve a}

Dirichlet Problem for the Laplace Operator in a Plane Sector, Differential and integral equations, Vol 3, N 2, 1990, 323-334

[8] _{R. Labbas, A. Medeghri, B. K. Sadallah; On a Parabolic}

Equation in a Triangular Domain. Applied Mathematics and Computation, 130, 2002, 511-523.

[9] _{P. Grisvard; Spazi di Tracce e Applicazioni, Rendiconti}

di Matematica (4), vol, 5 VI, 1972, 657-729.

[10] _{P. Grisvard; Probl`emes aux limites dans des domaines}

avec points de rebroussement. Annales de la Faculté des sciences de Toulouse : Mathématiques, Sér. 6, 4 no. 3,1995, 561-578

[11] _{M. Haase, The Functional Calculus for Sectorial}

Oper-ators and Similarity Methods, Thesis, Universitat Ulm, Germany, 2003.

[12] _{H. Hammou, R. Labbas, S. Maingot and A. Medeghri;}

Nonlocal General Boundary Value Problems of Elliptic Type in Lp Cases, Mediterr. J. Math. 13, N 4, 2016 1669-1683.

[13] _{N. M. Hung; N. T. Anh; Regularity of solutions of initial}

- boundary value problems for parabolic equations in domains with conical points, J. Differential Equations, 245, 2008, 1801-1818

[14] _{N. M. Hung, V. T. Luong, Unique solvability of initial}

boundary-value problems for hyperbolic systems in cylin-ders whose base is a cusp domain, Electron. J. Diff. Eqns., 138, 2008, 1-10.

[15] _{V. T. Luong; On the first initial boundary value problem}

for strongly hyperbolic systems in non-smooth cylinders, J. S. HNUE, 1(1) (2006), Vietnam.

[16] _{A. Lunardi, Analytic Semigroups and Optimal Regularity}

in Parabolic Problems,” Birkhauser, Basel, 1995.

[17] _{H. Triebel; Interpolation Theory, Function Spaces,}

Dif-ferential Operators, Amsterdam, North Holland, 1978.

[18] _{V. Usha, M. Mallika Arjunan; Existence results for}

neu-tral integro-differential systems with state-dependent de-lay in Banach spaces, Made-laya. J. Mat, Spec. Iss. 1, 2015, 314-325.

[19] _{A. D. Ventcel; On boundary conditions for}

multi-dimensional difusion processes. Theor. Probability Appl. 4, 1959, 164–177.

? ? ? ? ? ? ? ? ? ISSN(P):2319−3786

Malaya Journal of Matematik

On efficient abstract method for the study of Cauchy problem for a second order differential equation set on a singular cylindrical domain